N-body algorithms are the computational cornerstone for many problems in mathematical physics. In this talk, I will present (1) a brief history of N-body algorithms and their application to multiscale problems with an emphasis to fluid dynamics simulations; (2) give a brief overview of the most basic N-body algorithm, the Barnes-Hut method; and (3) conclude with applications of N-body algorithms to the simulation of Stokesian particulate flows.
Stokesian particulate flows are mixtures of a high viscosity Newtonian fluid and deformable capsules. Simulations of such flows require algorithms for infinite dimensional, highly stiff, nonlocal, and nonlinear dynamical systems. I will discuss some recent developments on numerical methods for such problems and I will report simulations with up to 260 million deformable capsules on the Oak Ridge National Laboratory's Cray XT5 "Jaguar" platform. The largest simulation amounts to 90 billion unknowns in space on 200,000 cores.